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I have a sphere (taken from http://www.texample.net/tikz/examples/, credits to Bartman):

 % Steradian cone in sphere
% Author: Bartman
documentclass[tikz,border=10pt]standalone
%%%<
usepackageverbatim
%%%>
begincomment
:Title: Steradian cone in sphere
:Tags: 3D;Angles;Intersections;Shadings;MMathematics;Geometry
:Author: Bartman
:Slug: steradian-cone-sphere

A graphical representation of a steradian.
It is the solid angle subtended at the center
of a unit sphere by a unit area on its surface. (Wikipedia)

Made by Bartman on
http://golatex.de/3d-kugel-in-tikz-t17380.html

The part of the cone is from http://tex.stackexchange.com/a/186109/213
endcomment
usepackagesansmath
usetikzlibraryshadings,intersections
begindocument
begintikzpicture[font = sansmath]
 coordinate (O) at (0,0);



 % ball background color
 shade[ball color = blue, opacity = 0.2] (0,0) circle [radius = 2cm];

 % cone
 beginscope
 defrx0.71% horizontal radius of the ellipse
 defry0.15% vertical radius of the ellipse
 defz0.725% distance from center of ellipse to origin

 path [name path = ellipse] (0,z) ellipse (rx and ry);
 path [name path = horizontal] (-rx,z-ry*ry/z)
 -- (rx,z-ry*ry/z);
 path [name intersections = of = ellipse and horizontal];


 endscope


 % ball
 draw (O) circle [radius=2cm];
 % label of ball center point
 filldraw (O) circle (1pt) node[below] $O$;

 % radius
 draw[densely dashed] (O) to [edge label = $1$] (-1.33,1.33);
 draw[densely dashed] (O) -- (1.33,1.33);

 % cut of ball surface
 draw[red, densely dashed] (-1.36,1.46) arc [start angle = 170, end angle = 10,
 x radius = 13.8mm, y radius = 3.6mm];
 draw[red] (-1.29,1.52) arc [start angle=-200, end angle = 20,
 x radius = 13.75mm, y radius = 3.15mm];

 % label of cut of ball surface
 draw (-1.2,2.2) -- (-0.23,1.1) node at (-1.37,2.37) $B$;
endtikzpicture
enddocument

I want to add an angle alpha like this:

How can I do this?

First, you have to name the coordinate for the edges of the angle. Here I use (x) and (y).

documentclass[tikz,border=10pt]standalone
usepackagesansmath
usetikzlibraryshadings,intersections,quotes,angles
begindocument
begintikzpicture[font = sansmath]
 coordinate (O) at (0,0);
 shade[ball color = blue, opacity = 0.2] (0,0) circle [radius = 2cm];
 beginscope
 defrx0.71% horizontal radius of the ellipse
 defry0.15% vertical radius of the ellipse
 defz0.725% distance from center of ellipse to origin
 path [name path = ellipse] (0,z) ellipse (rx and ry);
 path [name path = horizontal] (-rx,z-ry*ry/z)
 -- (rx,z-ry*ry/z);
 path [name intersections = of = ellipse and horizontal];
 endscope
 draw (O) circle [radius=2cm];
 filldraw (O) circle (1pt) node[below] $O$;
 draw[densely dashed] (O) to [edge label = $1$] (-1.33,1.33) coordinate (x);
 draw[densely dashed] (O) -- (1.33,1.33) coordinate (y);
 draw[red, densely dashed] (-1.36,1.46) arc [start angle = 170, end angle = 10,
 x radius = 13.8mm, y radius = 3.6mm];
 draw[red] (-1.29,1.52) arc [start angle=-200, end angle = 20,
 x radius = 13.75mm, y radius = 3.15mm];
 draw (-1.2,2.2) -- (-0.23,1.1) node at (-1.37,2.37) $B$;

 % Command for the angle
 pic[draw,->,angle radius=.5cm,angle eccentricity=1.3,"$alpha$"] angle=y--O--x;
endtikzpicture
enddocument

This is just a small addendum to Joule V's nice answer, which solved the main problem. By now there are IMHO much better tools available to draw such graphs.

1. 
   tikz-3dplot allows you to install orthographic projections, i.e. dial the view angles.


2. The 3d library allows you to switch to a plane to e.g. draw a latitude circle. So you no longer need to guess ellipses.


3. The angle of visibility, i.e. the angle at which the dashed lines turn in solid ones and vice versa has been compute e.g. here, so you do not need to guess this either.

documentclass[tikz,border=3.14mm]standalone
usepackagetikz-3dplot
usetikzlibrary3d,backgrounds,quotes,angles
begindocument
tdplotsetmaincoords8000
begintikzpicture[tdplot_main_coords]
 pgfmathsetmacroR2 % radius
 pgfmathsetmacromyang50 % latitude angle of the red circle
 coordinate (O) at (0,0,0);
 shade[ball color = blue, opacity = 0.2,tdplot_screen_coords] 
 (O) circle [radius = R*1cm];
 beginscope[canvas is xy plane at z=R*sin(myang),transform shape]
 % angVis from https://tex.stackexchange.com/a/49589/121799
 pgfmathsetmacroangVisatan(sin(myang)*cos(tdplotmaintheta)/sin(tdplotmaintheta))
 beginscope[on background layer]
 draw[red,dashed] (angVis:R*cos(myang)) arc (angVis:180-angVis:R*cos(myang));
 endscope 
 draw[red] (180-angVis:R*cos(myang)) arc (180-angVis:360+angVis:R*cos(myang));
 path (0:R*cos(myang)) coordinate (R) 
 (180:R*cos(myang)) coordinate (L);
 endscope
 beginscope[on background layer]
 draw[dashed] (L) -- (O) node[midway,below] $L$ -- (R);
 fill (O) circle[radius=1pt] node[below] $O$;
 pic[draw,-latex,angle radius=.5cm,angle eccentricity=1.3,"$alpha$"] 
 angle=R--O--L;
 endscope 
endtikzpicture
enddocument

The following animation shows that you can dial view and latitude as you wish.

documentclass[tikz,border=3.14mm]standalone
usepackagetikz-3dplot
usetikzlibrary3d,backgrounds,quotes,angles
begindocument
foreach Angle in 5,15,...,355
tdplotsetmaincoords70+cos(Angle)00
begintikzpicture[tdplot_main_coords]
 pgfmathsetmacroR2 % radius
 pgfmathsetmacromyang40+15*sin(2*Angle) % latitude angle of the red circle
 coordinate (O) at (0,0,0);
 shade[ball color = blue, opacity = 0.2,tdplot_screen_coords] 
 (O) circle [radius = R*1cm];
 beginscope[canvas is xy plane at z=R*sin(myang),transform shape]
 % angVis from https://tex.stackexchange.com/a/49589/121799
 pgfmathsetmacroangVisatan(sin(myang)*cos(tdplotmaintheta)/sin(tdplotmaintheta))
 beginscope[on background layer]
 draw[red,dashed] (angVis:R*cos(myang)) arc (angVis:180-angVis:R*cos(myang));
 endscope 
 draw[red] (180-angVis:R*cos(myang)) arc (180-angVis:360+angVis:R*cos(myang));
 path (0:R*cos(myang)) coordinate (R) 
 (180:R*cos(myang)) coordinate (L);
 endscope
 beginscope[on background layer]
 draw[dashed] (L) -- (O) node[midway,below] $L$ -- (R);
 fill (O) circle[radius=1pt] node[below] $O$;
 pic[draw,-latex,angle radius=.5cm,angle eccentricity=1.3,"$alpha$"] 
 angle=R--O--L;
 endscope 
endtikzpicture
enddocument